#### Answer

$x=2, y = -3 \text{ or } (2, -3)$

#### Work Step by Step

The system of equations is given by the following two equations:
$$\begin{cases}
\dfrac{1}{2}x+y&=-2\hspace{20pt} &(1)\\[3mm]
x-2y&=8 \hspace{20pt} &(2)
\end{cases}$$
$\text{Multiplying equation (1) by 2 so that the coefficients of y in the two equations are additive inverses gives:}$
$x+2y=-4\hspace{20pt} (1)$
$\text{Add the two equations to eliminate } y:$
\[
\begin{aligned} &x+2y=-4 \\[2mm]
&x-2y=8 \ \\[3mm]
\hline & 2x=4\\[2mm]
\end{aligned}
\]
Divide $2$ to both sides to obtain:
$x=2$
$\text{Substituting in equation (1) for }x=2$ gives:
$\dfrac{1}{2} (2)+y=-2$
$y=-2-1$
$y=-3$
Therefore, the solution is:
$\boxed{x=2 \hspace{20pt} y = -3}$